Digital devices for spectrum analysis

ABSTRACT

This invention relates to digital devices for spectral analysis. 
     Signal x(t) of spectrum X(f), sampled at frequency 1/T and encoded, is processed by a temporal digital filter having a width NT followed by one or more perfect digital resonators in parallel, each tuned to a specific predetermined frequency f p  and possessing two outputs, one being the &#34;sine&#34; output and the other being the &#34;cosine&#34; output on which the digital signals X(f p ) sin 2πf p  kT and X(f p ) cos 2πf p  kT are obtained after time NT, from which the value X 2  (f p ) can be formed by conventional digital processing. 
     Application to encoded multifrequency signal receivers used in telephone switching.

BACKGROUND OF THE INVENTION

(a) Field of the Invention

The present invention relates to methods and apparatus for measuring thepower or amplitude densities of the spectral components of a digitalsignal comprising a sequence of numbers x(nT), where T is the samplingperiod and n is the rank of the sample. The technique disclosed appliesto either signals naturally limited in time of interval ρ=NT, or signalsof unlimited duration previously subjected to temporal filtering ofinterval ρ.

(b) Discussion of the Prior Art

Known procedures for measuring the power spectral density of digitalsignals most often use methods which are based on either selectivefiltering, followed by quadratic integration, or on a calculation of thediscrete Fourier transform (DFT) of the signal to be analyzed. The majordisadvantage of these latter methods--apart from the complexity ofimplementation when N is large--is that they allow analysis of thespectrum only for frequencies which are multiples of 1/NT.

SUMMARY OF THE INVENTION

As a solution to these and other problems the instant inventioncomprises methods and apparatus for measuring the spectral densityspecific to a signal x(t), i.e. the autospectrum.

Methods and apparatus are also disclosed for measuring the interactionspectral density, i.e. the interspectrum, of two signals x(t) and x'(t).

Yet another embodiment of the invention discloses methods and apparatusfor recognizing and measuring the levels or amplitudes of sinusoidalfrequencies f₁, f₂ . . . f_(q) used in multifrequency signallingdevices, especially for signalling between automatic telephone exchanges(e.g. the so-called R₂ Code and SOCOTEL M.F. Code), or betweensubscribers and telephone exchanges (e.g. multifrequency Codes).

Among the advantages of the present invention are the following

(1) the length (NT) of the sequences to be processed can have any value;

(2) the analyzer disclosed allows the measurement of the spectraldensity associated with a frequency f, a group of frequencies f, or allthe frequencies in the range 0<f<1/2T;

(3) the frequencies, f are not necessarily special frequencies as in thecase of digital analyzers using the DFT method (f=r/NT, where 0≦r≦N-1);they may in fact, have any value;

(4) in the case of signals of finite duration, it is possible to refinethe analysis to any required extent; in other words, the frequencyresolving power threshold level Δf=1/τ, which limits the performance ofconventional analyzers, can be improved by increasing N, at least eachtime it is required to perform the measurement over a sufficiently longinterval τ;

(5) the analyzer disclosed is well suited for the automatic andprogrammed processing of signals.

The apparatus of the present invention uses algorithms which are basedon the theory of the z-transform, i.e. [z=exp(sT)], which is a specialcase of the Laplace transform the symbolic variable s applied totemporal functions sampled at the frequency F_(e) =1/T.

The methods used are an application of Plancherel's theorem which statesthat the Fourier transform, the Laplace transform or the z-transform ofthe convolution of two signals is equal to the product of theirindividual transforms, i.e. in the case of a Fourier transform, forexample:

    x(t).sub.* y(t)⃡X(f)·Y(f)

and reciprocally:

    x(t)·y(t)⃡X(f).sub.* Y(f)

For the sake of clarity, the sampled temporal functions and theirtransforms are represented by x*(t) and X*(f) in the followingdiscussion.

The theoretical considerations upon which the present invention is basedare developed in the following books:

(1) "Digital Processing of Signals" by B. Gold and Ch. M. Rader,published by McGraw-Hill, New York, (1969);

"Methodes et Techniques de Traitement du Signal et Applications auxMesures Physiques" by J. Max, edited by Masson (2nd edition, 1977).

And its mode of operation will be more fully understood from thefollowing detailed description when taken with the appended drawings inwhich:

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1a and 1b are a set of graphs showing the parallel evolution of atemporal signal and its frequency transform when being processed by adevice in accordance with the present invention;

FIGS. 2 and 3 are block schematic diagrams of perfect resonators for usein a device according to the present invention;

FIG. 4 is a block schematic diagram of an illustrative device formeasuring the spectral density specific to a signal x(t) and/or theinteraction spectral density of two signals x(t) and x'(t);

FIG. 5 is a block schematic diagram of another illustrative device formeasuring the power spectral density of a digital signal which may beused for the reception and recognition of the frequencies of an encodedmultifrequency digital signal; and

FIG. 6 is a block schematic diagram of another illustrative device formeasuring spectral component amplitudes which may be used for the samepurpose as the apparatus shown in FIG. 5.

As will be subsequently explained, the illustrative analyzing devicedisclosed and claimed herein includes:

(a) a temporal digital filter of width τ=NT comprising a gate--orlimiting window--of appropriate shape whose input is fed with the signalx(t) to be analyzed , which signal was previously sampled at thefrequency F_(e) =1/T and linearly quantified and encoded;

(b) one or more digital filters used sequentially, or simultaneously inparallel, each representing a perfect resonator tuned to a predeterminedfrequency f_(p) =f₁, f₂ . . . or f_(q), each frequency filter beingprovided with an input fed with the signal obtained from the temporaldigital filter and either one output providing, when k≧N, the signalX(f_(p)) sin 2πf_(p) kT or X(f_(p)) cos 2πf_(p) kT, as required, or twooutputs, known as "cosine" and "sine" outputs, producing the signalsy_(r) =X(f_(p)) cos 2πf_(p) kT and y_(s) =X(f_(p)) sin 2πf_(p) kT, whenk≧N, X(f_(p)) representing the module of the frequency component f_(p)of the signal x(t) to be analyzed.

Theoretically, each of the perfect resonators comprises essentially of arecursive digital filter having the following transfer function,expressed in terms of z: ##EQU1##

The pulse response h_(p) *(t) of such a filter is the inverse transformof H_(p) *(z), i.e. the unlimited series h_(p) *(kT)=exp(j2πf_(p) kT),with k≧0.

In order to state the principal operating main characteristic of theinstant invention in as precise a manner as possible, it may be saidthat the instant invention produces the convolute of signal e*(t)produced by the temporal digital filter and of h_(p) *(kT) to producethe temporal signals required on the resonator outputs:

    y.sub.p *(kT)=e*(t).sub.* h.sub.p *(kT)

Representing the z-transform of y_(p) *(kT) by Y_(p) *(z), and thez-transform of e*(t) by E*(z), Plancherel's theorem gives:

    Y.sub.p *(z)=E*(z)·H.sub.p *(z)

According to one characteristic aspect of the present invention,relating specifically to devices for measuring the spectral density ofone signal x(t), each of the perfect resonators is provided with twooutputs (e.g. "cosine" and "sine") associated with two digitalmultipliers producing X² (f_(p)) sin² 2πf_(p) kT and X² (f_(p)) cos²2πf_(p) kT for k≧N, the multiplier outputs being connected to an adderwhich produces X² (f_(p)).

According to another characteristic aspect of the invention, specific todevices for measuring the interaction spectral density of two signalsx(t) and x'(t), the latter are first individually processed aftersampling by two groups of identical circuits, each comprising of:

(a) a temporal digital filter of width τ=NT;

(b) one or more frequency digital filters behaving as perfect resonatorsand tuned to frequencies f_(p) (f₁, f₂ . . . f_(q)) used eithersequentially or simultaneously in parallel, but each possessing both"consine" and "sine" outputs.

The following signals are obtained for k≧N at the outputs of a perfectresonator of the first group processing x(t):

    y.sub.r *=X(f.sub.p) cos 2πf.sub.p kT and y.sub.s *=X(fp) sin 2πf.sub.p kT,

while the following signals are obtained for k≧N at the outputs of theconjugate perfect resonator of the second group processing x'(t):

    (y.sub.r *)'=X'(f.sub.p) cos (2πf.sub.p kT-φ)

    and (y.sub.s *)'=X'(f.sub.p) sin (2πf.sub.p kT-φ),

where X_(p) f) and X'(f_(p)) are the moduli of the frequency componentsf_(p) in signals x(t) and x'(t) and φ is the phase difference betweenthese components.

As will be explained, four digital multipliers are associated with eachpair of conjugate resonators, each digital multiplier having two inputsfed with the following:

the first multiplier: y_(r) * and (y_(r) *)',

the second multiplier: y_(s) * and (y_(s) *)',

the third multiplier: y_(s) * and (y_(r) *)',

and the fourth multiplier: y_(r) * and (y_(s) *)'.

The outputs of the first and second multipliers are associated with adigital adder, producing:

    R.sub.p =y.sub.r *·(y.sub.r *)'+y.sub.s *·(y.sub.s *)'=X(f.sub.p)·X'(f.sub.p) cos φ.

The outputs of the third and fourth multipliers are associated with adigital subtractor, producing:

    I.sub.p =y.sub.r *·(y.sub.s *)'-y.sub.s *·(y.sub.r *)'=X(f.sub.p)·X'(f.sub.p) sin φ.

R_(p) and I_(p) are the values required; they respectively represent thereal or in-phase component and the imaginary or quadrature component ofthe interaction spectral density of x(t) and x'(t) at frequency f_(p).

The analyzing devices in accordance with the present invention can beapplied, in principle, to any signal x(t) satisfying the followingconditions:

(1) the signal is bounded; ##EQU2## (3) the number of discontinuities inx(t), as well as of the number of maxima and minima, is finite;

(4) the spectrum of x(t) is limited, i.e. the transform X(f) is zerooutside of the interval -F_(M) to +F_(M).

In order to apply the invention, signal x(t)--if not already so--issampled at the frequency F_(e) =1/T, where F_(e) >2F_(M), and linearlyquantized and encoded to produce a digital series designated resin asx*(nT). The latter passed through a temporal digital filter of widthτ=NT comprising a limiting window which weights the samples as afunction of their rank n (0<n<N-1).

If the weighting law is represented by u(NT), the following signal isobtained at the temporal filter output:

    e*(nT)=x*(nT)·u(nT)  (0≦n≦N-1)

It may be noted in passing, that the use of a temporal digital filterallows the processing of signals x(t), such as x(t)=A_(p) sin 2πf_(p) t,which do not satisfy condition (3) above.

If E*(f), X*(f) and U(f) are the transforms of e*, x* and u,respectively, Plancherel's theorem gives:

    E*(f)=X*(f)*U(f)

Because of the importance of the limiting window when applying theinvention, it is useful to examine its structure in greater detail.

Function U(f) naturally depends on the interval τ=NT, but above all, onthe mathematical representation of u(nT), or of u(t) in continuous time.

If a simpler window is used, i.e. a square window defined by:

    u(t)=1 for -(τ/2≦ t≦(τ/2 )

    u(t)=0 for |t|≧(τ/2 )

the transform U(f) is written:

    U(f)=τ(sin πfτ/πfτ)

U(f) has a main peak of maximum amplitude G₀ =τ for f=0 and a totalwidth Δf₀ =2/τ, and secondary peaks of width 1/τ, alternately negativeand positive of slowly decreasing levels (-13 dB, -18 dB, etc.). In mostcases it is preferable to use a window producing acceptable secondarypeak levels in order to reduce interference between signals belonging todifferent bands of the spectrum.

It should also be noted that the variation of U(f), as a function of anyerror δf close to f=0, should not be too great. If the application tothe reception of multifrequency signals is considered, the constituentfrequencies of f_(p) are defined with a tolerance of ±Δf_(p), and sincethe corresponding perfect resonator is tuned to the nominal frequencyf_(p), an error occurs in the measurement of X(f_(p)).

The ideal spectrum should be rectangular, such that U (f)=τ for-1/τ≦f≦1/τ and U(f)=0 outside of these limits.

The corresponding temporal gate or window is theoretically defined bythe law: ##EQU3##

This is not possible, since it is of infinite duration.

It is seen, however, that u(t) has a main temporal peak of width 2τ anddecreasing-amplitude secondary peaks of width τ.

A certain number of digital gates providing a compromise between therectangular temporal gate and the theoretical rectangular spectrum gateare technically available.

In certain practical cases, the compromise consists of forming the timegate of total width τ by superpositioning k+1 truncated cosinusoidalgates of frequencies (k/τ) (k=0, 1, 2, etc.) and amplitudes α₀, α₁ . . .α_(k), where α₀ +α₁ +. . . +α_(k) =1.

A cosinusoidal gate of frequency k/τ has a spectrum with two main peakscentered on frequencies ±k/τ and whose amplitudes have the sign(-1)^(k). It may be considered that these cosinusoidal gate main peakscorrespond to the secondary peaks of the rectangular gate spectrum withequal amplitudes and opposite signs.

Remembering that the Fourier transform is a linear function, it is seenthat the resulting spectrum is the sum of the component gate spectra.

By adjusting the number k and the values of α₀, α₁ . . . α_(k), it isthus possible to obtain considerably reduced secondary peaks because ofthe compensations of signs between the peaks of the different temporalcomponents. This advantage is sometimes accompanied by widening of theresulting main peak.

Among gates of this type which are currently in use, the following maybe mentioned:

(a) The Hamming gate expressed by:

    u.sub.H (t)=0.54+0.46 cos 2π(t/τ)

where, -τ/2≦t≦τ/2

The main peak has a total width of 2Δf_(H) =4Δf₀ =4/τ and a maximumamplitude G_(H) =0.54τ. The secondary peak levels are at least 40 dBsmaller; and

(b) The Blackman gate expressed by:

    u.sub.B (t)=0.42+0.5 cos (2πt/τ) +0.08 cos (4πt/τ)

where -τ/2≦t≦τ/2.

The main peak has a total width of 2Δf_(B) =6Δf₀ =6/τ and a maximumamplitude G_(B) =0.42 τ. The secondary peak levels are at least 80 dBsmaller.

The time t is quantized and measured in multiples nT. The limiting gateor window opens for n=0 and closes for n=N-1.

This, therefore, produces:

    u.sub.H (nT)=0.54-0.46 cos 2π(n/N)   (0<n<N-1)

and

    u.sub.B (nT)=0.42-0.5 cos 2π(n/N)+0.08 cos 4π(n/N)   (0<n<N-1)

The weight given to the amplitude of each sample, as a function of itsrank n, is determined by the above formulae.

When measuring the spectral levels for the nominal frequencies f_(i) andf_(j), the frequency error between the main peak centre-frequency andeither edge of infinite attenuation must not exceed Δf_(ij) =|f_(i)-f_(j) |. In other words, in the case of a Hamming gate: τ_(H)=NT≧(2/Δf_(ij)), and in the case of a Blackman gate: τ_(B)=NT≧3/Δf_(ij).

Moreover, if frequencies f_(i) and f_(j) of the measured spectrum aredefined with a tolerance of ±Δf_(p), the minimum widths of the gates areagain increased and become: ##EQU4##

In the case of reception of multifrequency signals, using the SOCOTELM.F. code as an example, the frequencies are spaced 200 Hz apart andeach is defined with a tolerance of ±20 Hz. This therefore gives:

τ_(H) '≧11 ms

τ_(B) '≧16.5 ms

i.e. with T=0.125 ms, respectively:

N'=88

and N'=132.

If these results are considered, it is seen that the temporal gateduration τ for a spectrum of total given width of 2 ΔF increases as thetolerated secondary peak levels decrease.

It is, therefore, possible to write: ##EQU5##

There are many other types of gates, such as those which can besynthesized from the required spectrum shape. In particular, mention maybe made of the transformed gates of a Tchebyscheff filter lying withinthe following shape in which the positive values of f (half-spectrum)only are considered:

A_(max) dB in the pass-band Δf_(P)

A_(min) dB in the attenuated band beyond Δf_(ij) -Δf_(p).

In the case of application to the SOCOTEL M.F. system with:

A_(max) =0.2 dB within Δf_(p) =20 Hz

A_(min) =38 db beyond f=180 Hz (ΔF)

the value obtained is τ=9 ms.

The formula ΔF=α/τ gives α=1.6.

With respect to the width τ, it is seen that the Tchebyscheff gate liesbetween the rectangular gate (α=1) and the Hamming gate. The sameapplies to the power lost in the secondary peaks, since the Tchebyschefffilter possesses a large number of peaks having maximum levels of-A_(min) in the attenuated band, whilst the spectrum of the Hamming gatepossesses a limited number of secondary peaks of significant amplitude,the remaining peaks rapidly decreasing beyond the third.

Finally, the choice of gate depends on a compromise between the requiredaccuracy of spectral power measurement and the duration of measurement.If high accuracy is required, whilst avoiding the effect of interferencedue to secondary peaks, it is preferable to use a Blackman gate(α=approximately 3). If a short duration, is required however, it ispreferable to adopt a gate of the Tchebyscheff type (α=approximately1.5).

Theory makes it possible to represent the parallel evolution of digitalsignals and their transforms from x(t) to e*(nT) and X(f) to E*(f).

The detailed description of the invention which will now be given usesgraphical representations in preference to a mathematical developmentwhich, although more rigorous, tends to mask physical realities becauseof its rather complex symbolism.

Moreover, for the sake of clarity, we shall assume the use of a signalx(t) whose spectrum contains only two discrete frequencies f_(i) andf_(j), i.e.:

    x(t)=X.sub.i sin 2πf.sub.i t+X.sub.j sin (2πf.sub.j t-φ.sub.ij)

Extension to a signal containing a higher number of discrete frequenciesor a continuous-spectrum signal is simple.

In the drawings the right-hand column in FIG. 1a illustrates theevolutionary stages of signal x(t), whilst the left-hand columnillustrates those of the spectral transform x(t).

These parallel evolutions are analyzed as follows:

In Diagram 1-1, signal x(t), theoretically unlimited in time, is shownon the right. The spectrum X(f), consisting of four components ±f_(i)and ±f_(j) of amplitudes X_(i) /2 and X_(j) /2 is shown on the left.

In Diagram 1-2, the Dirac temporal train of period T, and expressed by##EQU6## is shown on the right. Its spectral transform, i.e. the Diracfrequency series of period F_(e) =1/T and expressed by ##EQU7## is seenon the left, but limited because of drawing limitations to the threecomponents corresponding to λ=-1, λ=0 and λ=1.

In Diagram 1-3, the simple product of x(t) and the Dirac temporal train,i.e. x*(nT), is shown on the right. The modulus of X*(f), the transformof x*(nT)--which is the product of convolution of X(f) and the Diracfrequency series--is shown on the left. This is a group of line spectra,each spectrum possessing four lines of frequencies λF_(e) ±f_(i) andλF_(e) ±f_(j) and of amplitudes X_(i) /2T (for f_(i)) and X_(j) /2T (forf_(j)).

In Diagram 1-4, the limiting gate of law u(t) for 0≦t≦τ is shown on theright, whilst its continuous spectrum U(f) of maximum amplitude G forf=0 is seen on the left. Since amplitudes X_(i) and X_(j) assigned tofrequencies f_(i) and f_(j) should finally be obtained in the remainderof the spectral transformations, the samples u(nT) defining the limitinggate are all multiplied by 2T/G.

In Diagram 1-5, the simple product of e*(nT) of u(nT) and x*(nT) isshown on the right, whilst the modulus of the product of convolutione*(f) of u(f) and X*(f) is seen on the left. In this case, this is amultiband spectrum derived from the line spectrum of diagram 1-3 inwhich, neglecting the presence of the secondary peaks of spectrum U(f),each line is surrounded by a specific band having a total width of 2α/τ.The amplitudes corresponding to frequencies λF_(e) ±f_(i) and λF_(e)±f_(j) are X_(i) and X_(j) respectively if the characteristics of thegate are those indicated above.

The following Diagrams 1-6, 1-7, etc., shown in FIG. 1b will beexplained in the remainder of the detailed description.

The signal processed by the perfect resonator or resonators tuned tofrequency f_(p) is 2T/Ge*(nT).

A resonator of theoretically perfect structure is shown in FIG. 2. Itcomprises of:

a digital adder 2-1 with two inputs, one input being fed with the signal2T/Ge*(nT), and one output;

a delay circuit 2-2 producing the unit delay T connected between theoutput of adder 2-1 and the input of a multiplier 2-3 having amultiplication factor of exp(j2πf_(p)), whose output is connected to theother input of adder 2-1.

The required signal (y₄ *+jy_(s) *) is obtained on the "complex" outputof adder 2-1.

The transfer function of the perfect resonator illustrated by FIG. 2 iswritten as follows: ##EQU8## with the following pulse response:

    h.sub.p *(kT)=exp(j2πf.sub.p kT)= cos 2πf.sub.p kT+j sin 2πf.sub.p kT

The multiplication factor exp(j2πf_(p) T) has no physical existence.Moreover, it is necessary to obtain y_(r) * and y_(s) * simultaneously.It is for this reason that the theoretical perfect resonator shown inFIG. 2 is replaced, in practice, by the perfect resonator shown in FIG.3, which is functionally identical. Moreoever, it is seen that writingH_(p) *(z) with real coefficients in the denominator is as follows:##EQU9## and the perfect resonator shown in FIG. 3 comprises abiquadratic recursive digital filter whose denominator possesses twoconjugate roots: exp (j2πf_(p) T) and exp(-j2πf_(p) T).

This second resonator includes of:

a first adder 3-1 having three inputs and one output, and a second adder3-2 having two inputs and one output C ("Cosine" Output), the output ofadder 3-1 being connected to one of the two inputs of adder 3-2;

first and second delay circuits 3-3 and 3-4 each producing unit delaysT, said delay circuits being connected in series;

a first digital multiplier 3-5 of factor 2 cos 2πf_(p) T connectedbetween the output of delay circuit 3-3 and the first of the threeinputs of adder 3-1;

a second digital multiplier 3-6 of factor -1 connected between theoutput of delay circuit 3-4 and the second of the three inputs of adder3-1;

a third digital multiplier 3-7 of factor sin 2πf_(p) T connected betweenthe output of delay circuit 3-3 and an output S ("sine" output) on whichthe imaginary part y_(s) * of the required signal is obtained;

a fourth digital multiplier 3-8 of factor - cos 2πf_(p) T connectedbetween the output of delay circuit 3-3 and the second input of adder3-2.

Adders 3-1 and 3-2 may comprise Texas Instruments 74LS32 Quod 2-inputPositive Or-gates. Digital multipliers 3-5 to 3-8 may comprise TexasInstruments 74LS00 Quod 2-input positive Nand-gates. The delay circuits3-3 and 3-4 may comprise physical inductive delay lines or TexasInstruments 74LS74 Dual Positive Edge Triggered Flip-Flops.

Incoming signal (2T/Ge*(nT) is applied to the third input of adder 3-1.

The real part y_(r) * of the required signal is obtained on output C("cosine" output).

Examination of the parallel changes to signal e*(nT) and its transformE*(f) may be continued by referring to FIG. 1b.

The following diagrams are seen in succession in FIG. 1b, which forconvenience starts with Diagram 1-5 of FIG. 1a:

In Diagram 1-6, the line spectrum on the left defined by: ##EQU10## andthe inverse transform cos 2πf_(p) kT on the right. In Diagram 1-7, theline spectrum on the left defined by: ##EQU11## and the inversetransform sin 2πf_(p) kT on the right. In Diagram 1-8, on the left thesimple product of Y_(r) *(f) of E*(f) shown in 1-5 and the line spectrumof 1-6, and on the right the convolution product of the inversetransforms: y_(r) *=|X(f_(p))|. cos 2πf_(p) kT (with k>N) when f_(p)=f_(j).

In Diagram 1-9, on the left the simple product of Y_(s) *(f) of E*(f)shown in 1-5 and the line spectrum of 1-7, and on the right theconvolution product of the inverse transforms: y_(s) *=|X(f_(p))|. sin2πf_(p) kT (with k>N) when f_(p) =f_(j). Products y_(r) * and y_(s) *,which are the required values, appear on outputs C and S respectively ofthe perfect resonator shown in FIG. 3.

FIG. 4 is the diagram of a first illustrative spectral density deviceaccording to the invention which allows the simultaneous measurement ofthe spectral density specific to a signal x(t) for two frequencies f₁and f₂ and/or the interaction spectral density of two signals x(t) andx'(t) for a single frequency f₁.

It is assumed that both signals have been previously sampled at thefrequency F=1/T, quantized and encoded.

The circuit diagram shown in FIG. 4 is divided into two halves by theline A-B. Part "A" shows the components of the circuits processingx*(nT) for frequency f₁ and part "B" shows those completely processingx'*(nT) for the same frequency f₁, or partly processing x*(nT) foranother frequency f₂.

A temporal digital filter e.g. by multiplier 4-1a in part "a" multipliesthe samples x*(nT) by the amplitudes (2T/G)u(nT) of a limiting gatestored in a read-only memory 4-2a, u(nT) being zero outside of theinterval 0≦n≦N-1 and G representing the maximum amplitude of thetransform U*(f) of u(nT).

Signals e*(nT) produced by multiplier 4-1a enter a read-write memory4-3a comprising, for example, a looped shift register which stores Nsamples of e*(nT0. These N samples are processed by a perfect resonator4-4a of transfer function H₁ *(z) tuned to frequency f₁ and which has,for example, the structure shown in FIG. 3. After a time τ=NT, therequired signals X(f₁) cos 2πf_(a) kT and X(f₁) sin 2πf_(a) kT areobtained on outputs C_(a) and S_(a) of resonator 4-4a.

The right side of part "A" shows two switches 4-5a and 4-6a each havingtwo contacts a and b. Outputs C_(a) and S_(a) are connected to contactsa of switches 4-5a and 4-6a, respectively.

The two inputs of a multiplier 4-6a are respectively connected to outputC_(a) and switch 4-5a. Similarly, the two inputs of a multiplier 4-8aare respectively connected to S_(a) and to a switch 4-6a. The outputs ofmultipliers 4-7a and 4-8a are added by an adder 4-9a. If switches 4-5aand 4-6a are in position a, the required value |X(f₁)|² appears on theoutput of adder 4-9a.

Circuit elements 4-1b, 4-2b and 4-3b in part "B", after having thenecessary changes made thereto, are similar to the corresponding circuitelements 4-1a, 4-2a and 4-3a. The output of 4-3b read-write memory isconnected to contact b of a 2-position switch 4-10, whose other contact,a, is connected to the output of 4-3a read-write memory. 4-10 switch isconnected to the input of another perfect resonator 4-4b of transferfunction H₂ *(z) which is tuned:

(a) either to the same frequency f₁ as resonator 4-4a when the device isused for measuring the interaction spectral density of signals x(t) andx'(t), switch 4-10 being in position b for this purpose; or

(b) to frequency f₂ when the apparatus is intended for simultaneouslymeasuring the spectral density specific to x(t) at two frequencies f₁and f₂, switch 4-10 then being in position a.

After a time τ=NT, outputs C_(b) and S_(b) of resonator 4-4b produce therequired signals:

X(f₂) cos 2πf₂ kT and X(f₂) sin 2πf₂ kT,

if 4-10 is in position a; and

X'(f₁) cos 2πf₁ kT and X'(f₁) sin 2πf₁ kT,

if 4-10 is in position b.

The right side of part "b" shows two switches 4-5b and 4-6b, each havingtwo contacts a and b. Outputs C_(f) and S_(f) are connected to contactsa of switches 4-5b and 4-6 respectively.

The two inputs of a multiplier 4-7b are respectively connected C_(b) andswitch 4-5b. Similarly, the two inputs of a multiplier 4-8b respectivelyare connected S_(b) and to switch 4-6b.

The outputs of multipliers 4-7b and 4-8b add (or subtract) in aswitchable device 4-9b, which operates as an adder if switches 4-5a,4-6a, 4-5b, 4-6b and 4-10 are in position a, or as a subtractor if theyare in position b. In the first case, the required value |X(f₂)|²appears on the output of adder 4-9b.

In order to measure the interaction spectral density, this measurementbeing made when the five switches are in position b, the followingconnections are made between the components shown in the right side ofparts "a" and "b":

C_(a) and contact b of 4-6b;

S_(a) and contact b of 4-5b;

C_(b) and contact b of 4-5a;

S_(b) and contact b of 4-6a.

If φ is the phase difference between the components of x(t) and x'(t)having a frequency f₁, the pairs of signals on the inputs of multipliers4-7a, 4-8a, 4-7b and 4-8b are as follows under these conditions:

on 4-7a: X(f₁) cos 2πf₁ kT and X'(f₁) cos (2πf₁ kT-φ);

on 4-8a: X(f₁) sin 2πf₁ kT and X'(f₁) sin (2πf₁ kT-φ);

on 4-7b: X(f₁) sin 2πf₁ kT and X'(f₁) cos (2πf₁ kT-φ);

on 4-8b: X(f₁) cos 2πf₁ kT and X'(f₁) sin (2πf₁ kT-φ);

The following is produced on the output of adder 4-9a:

X(f₁).X'(f₁) cos φ

whilst the following is obtained on the output of circuit element 4-9bwhich operates in this case as a subtractor:

X(f₁).X'(f₁) sin φ,

which are the two required quantities.

The coefficients of the perfect resonators 4-4a and 4-4b correspondingto frequency f₁ or to the pair of frequencies f₁ and f₂ can be stored ina read-only memory 4-11.

It is necessary to make sure that the delay circuits 3-3 and 3-4 of theperfect resonators 4-4a and 4-4b are reset to zero following the entryof N samples into the read-write memories 4-3a and 4-3b, in order to beable to process a further sequence of N samples either for the samefrequencies f₁ and f₂ or for other frequencies whose coefficientsapplicable to resonators 4-4a and 4-4b are obtained from the read-onlymemory 4-11. This zero reset may be performed by means not shown in FIG.4, using for example, the read-write memories 4-3a and 4-3b themselves.

We shall now give an example of the application of the analysis deviceaccording to the present invention to the reception of encodedmultifrequency digital signals.

Telephone exchanges use multifrequency signalling codes for signallingbetween exchanges, or for signalling between a subscriber's telephoneand an exchange. These codes typical comprise combinations of twofrequencies out of q frequencies (q being 6, 7 or 8).

The principle of operation of most multifrequency receivers is based onthe use of selective attenuating filters centered on the frequencies tobe detected, or on the calculation of the discrete Fourier transform ofa series of N samples of the input signal to be decoded.

Both these methods have disadvantages. The first requires the use ofvery selective and relatively costly narrow-band filters. The second issuitable for signalling codes whose frequencies are in arithmeticprogression, but cannot be used in the case of the codes, which consistsof frequencies in geometric progression.

FIG. 5 is the block diagram of an illustrative frequency analyzeraccording to the invention for measuring the power level assigned toeach of the q (q=8 in the example given) predetermined frequencies.

Signal x(t) is of the form:

x(t)=X_(i) sin 2πf_(i) t+X_(j) sin (2πf_(j) t-φ_(ij))+ various residualspurious signals in which f₁ and f_(j) are two of the 8 codefrequencies, X_(i) and X_(j) are the amplitudes and φ_(ij) is the phasedifference between the sinusoids.

After sampling, with possible quantizing and encoding, the resultingsignal x*(nT) is applied to a temporal digital filter comprising amultiplier 5-1, which multiples x*(nT) by the amplitudes (2T/G)u(nT) ofa limiting gate contained in a read-only memory 5-2, u(nT) being zerooutside of the interval 0<n<N-1 and G representing the maximum amplitudeof the transform U*(f) of u(nT).

The signals e*(nT) obtained from multiplier 5-1 are processed by a groupof 8 perfect resonators 5-3 each having two outputs C and S anddesignated by 5-31 to 5-38. These resonators are in parallel and each istuned to one of the eight frequencies of the code being studied. Aread-only memory 5-4 contains the coefficient values 2 cos 2πf_(p) T,-cos 2πf_(p) T and sin 2πf_(p) T for all the frequencies of the variouscodes which the analyzer is intended to receive. These coefficients aretransmitted to the perfect resonators 5-31 to 5-38 by connections 5-41to 5-48.

The outputs C and S of each perfect resonator are respectively connectedto multipliers 5-51c and 5-51s, 5-52c and 5-52s, . . . 5-58c and 5-58s.

The outputs of each pair of multipliers are added in adders 5-61, 5-62,. . . 5-68.

The required quantities X_(i) ² and X_(j) ² are obtained on the outputsof adders 5-6i and 5-6j after a time τ=NT following the appearance ofthe first significant sample on the input of multiplier 5-1.

The outputs of the other adders may also produce level indicationsperceptibly smaller than X_(i) ² and X_(j) ². These are, in particular,various spurious signals associated with harmonics nf_(i) and n'f_(j) ofthe code signals, since frequencies nf_(i) ±mF_(e) and n'f_(j) ±m'F_(e)appear in the sampled spectrum and which may be equal to anotherfrequency f_(k) of the code, or neighbouring frequencies. Because of thedesign of the system, however, the levels of the spurious signalscompared with X_(j) ² and X_(i) ² cannot exceed those of harmonicsnf_(i) and n'f_(j) in the transmitted signal, as defined by thespecifications for the various multifrequency code systems.

A logical decision device 5-7 classifies the levels X_(i) ², X_(j) ²,etc., compares them, and decides if they belong to a code according tothe signalling specifition. Logical decision device may comprise aplurality of 74LSφφ Quod dual input Nand-gates.

According to the principle of the present invention, the measurement ofX_(i) ² and X_(j) ² is obtained after the time τ=NT. Conventional meansnot shown in FIG. 5 and possibly associated with logical decision device5-7, should be provided to return the delay circuits of the resonators5-31 to 5-38 concerned to their initial states when k≧N. The same meansshould also open the limiting gate which closes after the passage of Nsamples, i.e. after the time τ. The choice of the instant MT>NT forreturning to the initial state is left to the discretion of the user,since in principle X_(i) ² and X_(j) ² no longer vary after time NT.

We will now give a second example of the use of an analysis deviceaccording to the present invention for the reception of encodedmultifrequency digital signals.

The digital signals of the multifrequency codes mentioned in the firstexample have a sampling frequency F_(e) =1/T=8000 Hz greater than fourtimes the highest frequency of the code frequencies to be detected (1980Hz for the "R₂ " code). There are, therefore, more than four samples perperiod for representing the temporal signals of the code frequenciesdetected, and theoretical analysis shows that it is sufficient toobserve the phenomena on the "cosine" or "sine" outputs of the perfectresonators shown in FIG. 3.

FIG. 6 is the block diagram of another illustrative frequency analyzerfor measuring the relative amplitudes of the sinusoids which carry the qpredetermined code frequencies.

As in the first example, signal x(t) is of the form:

x(t)=X_(i) sin 2πf_(i) t+X_(j) sin (2πf_(j) t-φ_(ij))+various residualspurious signals and is processed by circuit components 6-1 and 6-2which are similar to the homologous circuit devices 5-1 and 5-2discussed with reference to FIG. 5.

The signals e*(nT) produced by multiplier 6-1 are processed by a groupof 8 substantially identical perfect resonators 6-31 to 6-38, shown inrectangle 6-3, each possessing a single output, the "sine" outputdesignated by S₁, S₂ . . . S₈. These resonators are in parallel, andeach is tuned to one of the 8 code frequencies being studied.

Inside the deliberately enlargened rectangle 6-31 are shown theresonator components. They differ from the arrangement shown in FIG. 3in that adder 3-2 and multiplier 3-8. Are not needed and, therefore areomitted.

The other resonators 6-32 to 6-38 are configured in the same manner asresonator 6-31.

A read-only memory 6-4 contains the values of the two coefficients 2 cos2πf_(p) T and sin 2πf_(p) T for all the frequencies of the various codeswhich the analyzer is intended to receive. These coefficients aretransmitted to perfect resonators 6-31 to 6-38 by connections 6-41 to6-48.

The description of the operation of the analyzer shown in FIG. 6, fromthe input x*(nT) to outputs S₁ to S₈, is similar to that already givenabove for the analyzer shown in FIG. 5.

Signal x(t) is of the form:

x(t)=X_(i) sin 2πf_(i) t+X_(j) sin (2πf_(j) t-φ_(ij))+various residualspurious signals. The following sampled signals are obtained on outputsS_(i) and S_(j) of resonators 6-3i and 6-3j:

    y.sub.i *=X.sub.i sin 2πf.sub.i kT

    and y.sub.j *=X.sub.j sin 2πf.sub.j kT

for k≧N, i.e. after a time τ=NT representing the total width of the timegate 6-2.

From k=0 (initialization) to k=N, the digital signals on outputs S_(i)and S_(j) are approximately represented by:

    y.sub.i *=ε.sub.i sin 2πf.sub.i kT

    y.sub.j *=ε.sub.j sin 2πf.sub.i kT

with ε_(i) and ε_(j) increasing exponentially from 0 to X_(i) and from 0to X_(j) as k varies from 0 to N.

The device shown in FIG. 6 are advantageously provided with means forextracting the digital values X_(i) and X_(j). These means, shown in theright side of the figure, are in fact digital amplitude detectors.

Each of the outputs S₁ to S₈ is connected to the input of a digitalrectifying circuit, such as rectifiers 6-51, 6-52, . . . 6-58, whichproduces on its output the absolute value of each sample of the inputsignal.

The 8 outputs of circuits rectifiers 6-51 to 6-58 are respectivelyconnected to the input of a plurality of low-pass digital filters 6-61to 6-68. The structure of these digital filters is shown inside thedeliberately enlargened rectangle filter 6-61. 6-61 comprises an adder awith two inputs, of which one is fed with the digital signals obtainedfrom rectifier 6-51, and an output; a delay circuit b of unit delay Tand connected between the output of a and the input of a multiplier chaving a coefficient A (A≦1) whose output is connected to the otherinput of adder a. These filters are of the first-order, recursive typehaving the following transfer function expressed in terms of z:##EQU12##

The pulse response of such a filter is the sampled exponential:##EQU13## and on the output of filter 6-6i, for example, the digitalsignal is equal to the convolution product of y_(i) * and d*(kT). Fork>N, this product tends to the required value X_(i) (multiplied by 2/π)with amplitude variations which are greater or lesser depending onwhether A is more or less distant from its maximum value 1, whichcorresponds to the stability threshold of filters 6-61 to 6-68. Each ofthe filters 6-61 to 6-68 may be followed by a circuit producing thearithmetic mean, such as averaging circuits 6-71, 6-72, . . . 6-78,thereby "smoothing" the digital signals produced by the said filtersaveraging circuits 6-71 to 6-78 may comprise a plurality of 74LS1934-bit up/down counters and a plurality of 74LS32 Quad 2-1 input positiveOR-gates.

A logical decision device 6-8 classifies the amplitudes X_(i), X_(j),etc., compares them and decides if they belong to a code according tospecification signalling. Logical Decision Device 6-8 may comprise aplurality of 74LSφφ Quod dual input Nand-gates.

The further A is different from 1, the longer the acquisition time ofX_(i) and X_(j). In all cases, it the acquisition time exceeds theminimum value τ=NT. The effective time of the measurement is thereforelonger than that required by the device shown in FIG. 5. The advantageof the present device is that it reduces the number of multiplicationsper unit time required for acquiring values X_(p).

Although the principles of the present invention are described above inrelation with specific practical examples, it should be cearlyunderstood that said descriptions are given as examples only and do notlimit the scope of the invention.

I claim:
 1. A digital device for analyzing the spectrum of a signal X(t), said signal having the spectral transform X(f), -Fe/2<f<Fe/2, and previously having been sampled at a frequency Fe=T, and linearly quantized and encoded to produce a digital series X* (nT), said device measuring the power spectral density of said signal and including:(a) a limiting gate of width τ=NT connected to the input of the device for processing the incoming signal X*(nT); (b) a perfect digital resonator, including first and second delay circuits each of delay T, connected to said limiting gate, said resonator having a first output S, known as the "sine" output and a second output "C" known as the "cosine" output; and (c) a read-only memory connected to said resonator, said memory storing q group of three coefficients, 2 cos 2π fpT, and sin 2π fpT, respectively, corresponding to the q frequencies of the analysis, characterized in that the device further comprises: (d) at least one read-write memory comprising a looped shift-register inserted between the output of said limiting gate and the input of said perfect resonator for storing N samples of said signal; and (e) means, connected to said first and second delay circuits in said perfect resonator, for resetting said delay circuits after the passage of the N samples stored in said read-write memory.
 2. A digital device according to claim 1 for measuring the interaction spectral density of two signals, x(t) and x'(t), having the spectral transforms X(f) and X'(f) respectively, for q frequencies f_(p), said device including:first and second substantially identical channels respectively processing signals x(t) and x'(t), each channel including, according to the teachings of claim 1, a limiting gate, a digital perfect resonator connected to a read-only memory storing the q groups of three coefficients corresponding to the q frequencies of the analysis, a read-write memory comprising a looped shift register inserted between the said limiting gate and the input of the said perfect resonator for storing N samples of said signal, and means for resetting the perfect resonator after the passage of the N samples stored in the read-write memory; characterized by: (f) first, second, third and fourth digital multipliers each having two inputs of the said first multiplier being connected to C and C', the inputs of said second multiplier being connected to S and S', the inputs of the third multiplier being connected to S and C', and the inputs of the fourth multiplier being connected to C and S', C and S respectively comprising the "cosine" and "sine" outputs of said first channel (processing x(t)) and C' and S' respectively comprising the "cosine" and "sine" outputs of said second channel (processing x'(t)); (g) an adder connected to the outputs of said first and second multipliers the output of said adder producing the in-phase part R_(p) of the interaction spectral density at frequency f_(p) ; (h) a subtractor connected to the outputs of said third and fourth multipliers, the output of said subtractor producing the quadrature part I_(p) of the interaction spectral density at frequency f_(p).
 3. A digital device in accordance with claim 1 for receiving encoded multifrequency signals each comprising the sampled, quantized and linearly encoded sum of several sinusoids of the form X(f_(p)) sin 2π f_(p) T, whose frequencies f_(p), defining a signalling code, are selected out of q predetermined frequencies, said device including q perfect digital resonators connected in parallel, each having first and second delay circuits and being tuned to one of the frequencies f₁, f₂ . . . f_(q) of the code, the "sine" output of each resonator being connected to a digital rectifier (6-51) thence to a digital low-pass filter (6-6) whose output produces the amplitude X(f_(p)) and a decision logic circuit (6-8) connected to the output of the low-pass filter which receives, classifies and compares the acquired values X(f_(p)), characterized in that said device further includes:(i) means connected to and functionally associated with said decision logic circuit, for resetting the first and second delay circuits of the perfect resonators concerned, and for reopening said limiting gate at the instant MT after time NT.
 4. A digital device, in accordance with claim 1, for receiving encoded multifrequency signals, said device including q perfect digital resonators (5-31) connected in parallel, each having first and second delay circuits and being tuned to a frequency f₁, f₂, . . . f_(q) of the code; q digital squaring circuits (5-51S) each having an input connected to output S of a corresponding one of the perfect digital resonators, q digital squaring circuits (5-51c) each having an input connected to output C of a corresponding one of the perfect digital resonators, q 2-input adders (5-61) connected to the outputs of the corresponding digital squaring circuits associated with the same perfect resonator, and a decision logic circuit connected to the outputs of said q adders which receives, classifies and compares the values X² (f_(p)) acquired on the outputs of the q adders, characterized in that said device further includes:(j) means connected to and functionally associated with said decision logic circuit, for resetting the first and second delay circuits of the perfect resonators concerned, and for reopening said limiting gate at the instant MT after time NT.
 5. A digital device in accordance with claim 1 or 2 or 3 or 4, wherein said limiting gate of width T=NT has a weighting curve u(nT), symmetrical about N/2T and a maximum amplitude of 1 for NT/2, and zero outside of the interval 0 to (N-1)T, characterized in that said limiting gate comprises at least one multiplier (4-1) which forms the product of the amplitude of each sample, according to its rank, and the quantities (1/Ku) (nT), said quantities being stored in a read-only memory (4-2), K being equal to G/2T and G measuring the amplitude of the spectral transform of u(nT).
 6. A digital device in accordance with claim 3 or 4, characterized in that the width T=NT of said limiting gate is selected to be equal to, or greater than, ##EQU14## wherein a is a coefficient lying between 1 and 3, depending on the type of gate used, Δf_(ij) is the spacing between the closest pair of frequencies in the code considered, and Δf_(p) is the tolerance with respect to values f₁, f₂. . . f_(q) in the received signal frequencies. 